Problem 45 - A Complete Resource Book in Mathematics for JEE Main

Question

If for all \(x, y\) the function \(f\) is defined by \(f(x)+f(y)+\) \(f(x) \cdot f(y)=1\) and \(f(x)>0\), then (A) \(f^{\prime}(x)\) does not exist (B) \(f^{\prime}(x)=0\) for all \(x\) (C) \(f^{\prime}(0)

Step-by-Step Solution

Answer

(D) None of these.

Step 1: Rewrite the functional equation

Given the functional equation is \(f(x) + f(y) + f(x) \cdot f(y) = 1\). Let's rewrite this equation as: \( (f(x) + 1)(f(y) + 1) = 2\). This suggests a possible form for \(f(x)\).

Step 2: Identify the form of \(f(x)\)

Analyzing the rewritten equation, we can set \( f(x) + 1 = \frac{2}{k(x)} \) and \( f(y) + 1 = k(x) \) which implies \( f(x) = \frac{2}{k(x)} - 1 \). Similarly, for \(f(y)\), equate both equations to recognize similarity: \( f(x) = \frac{1}{f(y)}\). Therefore, the form of \(f(x)\) is consistent with \( f(x) = \frac{1}{x} \).

Step 3: Check the validity and conditions

Substituting \(f(x)=\frac{1}{x}\) into the equation \( f(x) + f(y) + f(x) f(y) = 1 \), we check if it holds true. Rearrange it: \( \frac{1}{x} + \frac{1}{y} + \frac{1}{xy} = 1 \), which is a valid solution as it implies \( x+y+1=xy \). Thus, for \( x,y > 1 \), the equation holds true.

Step 4: Determine the derivative

If \(f(x) = \frac{1}{x}\), the derivative, \(f'(x) \), is obtained by applying the derivative formula of \(\frac{1}{x}\), which is \(-\frac{1}{x^2}\).

Step 5: Evaluate all options

With \(f'(x) = -\frac{1}{x^2}\), compute: 1. \(f'(x)\) does exist, so (A) is incorrect.2. \(f'(x)\) is not 0, as \(-\frac{1}{x^2} e 0\) for \( x > 0 \), so (B) is incorrect.3. For \(x < 1\), \(f'(0) < f'(1) \) is impossible since these derivatives are not valid in original condition (functions undefined or zero). Results do not match (C).4. Therefore, "None of these" must be the answer for (D).

Key Concepts

Derivative Calculus

Derivative calculus is a fundamental component of calculus focused on finding the rate at which a function is changing at any given point. In this exercise, we tell that the function is given by \(f(x) = \frac{1}{x}\). Derivatives help us understand how functions behave and change.

### Calculating the DerivativeFor \( f(x) = \frac{1}{x} \), applying the standard differentiation formula reveals \( f'(x) = -\frac{1}{x^2} \). This tells us the slope of the tangent line to the curve at any point \(x\).

• Negative Sign: Indicates a negative slope, meaning the function decreases as \(x\) increases.
• Power Rule: The formula \(f(x) = x^{-n}\) differentiates to \(-nx^{-n-1}\).

The derivative's form is consistent with the expectations for this reciprocal function. Knowing the derivative helps in understanding how steeply the function decreases with increasing \(x\).

Analyzing options, we determine that options A and B are incorrect, as they suggest the derivative is either non-existent or zero, neither of which aligns with the actual derivative function.

Mathematical Analysis

Mathematical analysis involves a collection of tools and concepts to comprehend the behavior and properties of functions. In the scope of this exercise, one of the critical components was identifying the functional relationship from the given equation.

### Analyzing FunctionsInitially, we started with the equation \( f(x) + f(y) + f(x) \cdot f(y) = 1 \). Through manipulation, the expression becomes \((f(x) + 1)(f(y) + 1) = 2\). This change in representation provides insight that aids in unraveling the structure, hinting towards solutions like \( f(x) = \frac{1}{x} \).

• Equation Rewriting: Simplifying expressions helps in identifying underlying patterns or forms.
• Consistency: Ensuring the derived forms consistently satisfy original equations is vital.

Understanding the analysis ensures that the function reflects expected behavior and that its properties align with given conditions. This approach proves beneficial in making accurate conclusions regarding the options presented in the exercise.

Problem Solving in Mathematics

Problem solving in mathematics requires understanding the problem, formulating strategies, and logically deriving solutions. The process shown in this exercise involves multiple steps that lead to the identification of a valid function.

### Steps in Problem Solving1. **Understand the Problem:** First step involved comprehending the functional equation and exploring its implications.2. **Rewrite and Analyze:** Modify the equation to make it more manageable, which in this case revealed potential forms.3. **Check Validity:** Substituting back to ensure correctness and adherence to conditions imposed by the problem.

• Experimentation: Trying different representations helps uncover correct pathways.
• Logical Evaluation: Confirm that results adhere to initial conditions and constraints.

Through these steps, we figured out that the function \(f(x) = \frac{1}{x}\) holds true under the given conditions and verified that none of the proposed derivative conditions in options A, B, or C matched correct conclusions. This problem solving process empowers us to identify the best solutions efficiently.